To find the perimeter of a triangle with vertices of #(1,2)#, #(3,−4)# and #(−4,5)#, we have to first find distance between each pair of points, which will give length of sides. For this we use distance formula between two points #(x_1,y_1)# and #(x_2,y_2)# is #sqrt((x_2-x_1)^2+(y_2-y_1)^2)#. Hence if lengths of sides are #L_1,L_2,L_3#, these are as follows:

#L_1=sqrt((3-1)^2+((-4)-(2))^2)=sqrt(2^2+(-6)^2)=sqrt(4+36)=sqrt40=2sqrt10=6.325#

#L_2=sqrt((-4-(3))^2+(5-(-4))^2)=sqrt((-7)^2+9^2)=sqrt(49+81)=sqrt130=11.402#

#L_3=sqrt((-4-1)^2+(5-2)^2)=sqrt((-5)^2+3^2)=sqrt(25+9)=sqrt34=5.831#

Hence Perimeter is #6.325+11.402+5.831=23.558#